## Thursday, September 15, 2016

### #Alg2Chat Rich Problems Part 2

So in Part 1 in exploring the topic of Rich Problems, I outlined typical characteristics of Rich Problems. They are non-routine in that they are not typical practice of a skill or algorithm; they have little scaffolding; there are multiple entry points; and students of varying skill levels can benefit from Rich Problems. Rich Problems have natural extensions - one question leads to another. They are difficult and interesting at the same time. Rich Problems are revealing! They give insight to teachers about students' understanding. They help students experience the essence of mathematics.

So if you are like me ... you want to see some examples of Rich Problems.  Examples help us think about our own work and how we might tweak a question to make it more open-ended, more accessible to students and yet keep it challenging!

I did a Google Search and found a few problems I think would fit nicely in Algebra 2, First Semester curriculum, and could potentially be worthy of discussion!  Let me know what you think of these!

Systems of Equations ... From Illustrative Math ... Cash Box
Nola was selling tickets at the high school dance. At the end of the evening, she picked up the cash box and noticed a dollar lying on the floor next to it. She said,
I wonder whether the dollar belongs inside the cash box or not.
The price of tickets for the dance was 1 ticket for $5 (for individuals) or 2 tickets for$8 (for couples). She looked inside the cash box and found\$200 and ticket stubs for the 47 students in attendance. Does the dollar belong inside the cash box or not?
Make up an arithmetic sequence of six numbers: a0, a1, a2, a3, a4, and a5. Then write and solve the system of equations (feel free to use technology to help you solve these).
Compare your answer to others who used a different sequence. Come up with conjectures as to both what is happening and why.  Be ready to justify your conjectures and to share your work.

Quadratic Functions ... from Underground Mathematics

Given that two of the parabolas have the equations, y=x212x+27      and  can you find the equations of the other parabolas?  Can you recreate the diagram on Desmos?

Quadratic Functions ... from Underground Mathematics
When are coefficients of a quadratic equal to its roots?
The equation 𝑥^2 + 𝑎𝑥 + 𝑏 = 0, where 𝑎 and 𝑏 are different, has solutions 𝑥 = 𝑎 and   𝑥 = 𝑏.  How many such equations are there?

Non-Linear Systems of Equations ... from Riverpoint Advanced Mathematics Partnership-Algebra Project ... Intersections

Part 1: A line with slope 5 passes through the vertex of the parabola above. Does it intersect the parabola in another point (other than the vertex)? If so, find the point of intersection. If not, explain why.
Part 2: Think of all possible lines that pass through the vertex of the parabola shown above. Which lines intersect the parabola again at another point and which ones do not? Explain.
Part 3: Think of all possible lines with a slope of 5. Which of these lines intersect the parabola shown above? How many times? Explain.

There are others!  Certainly many of the Desmos Activity Builders are rich, worthwhile problems - as are 3-Act Math Tasks.

Do you have a favorite "Rich Problem?"  Do you have a favorite source for problems??  If so, please share in the comments!

Resources for problems include ...